Abstract
The Wasserstein diffusion is an Ornstein–Uhlenbeck type process on the set of all probability measures with the Wasserstein metric as intrinsic metric. Sturm and von Renesse constructed in [6] this process in the case of probability measures over the unit interval using Dirichlet form theory. An essential step in this construction is the closability of a certain gradient form, defined for smooth cylindrical test functions, in the space L 2 w.r.t. the entropic measure ℚβ. In this paper we will first give an alternative proof for this closability, avoiding the striking, but elaborate integration by parts formula for ℚβ used in [6]. Second, we give explicit conditions under which certain finite-dimensional particle approximations introduced in the paper [1] by Andres and von Renesse do converge in the resolvent sense to the Wasserstein diffusion, a question that was left open in the above cited paper.
Mathematics Subject Classification (2010). Primary: 58J65, Secondary: 47D07, 60J35, 60K35.
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References
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Stannat, W. (2013). Two Remarks on the Wasserstein Dirichlet Form. In: Dalang, R., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications VII. Progress in Probability, vol 67. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0545-2_12
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DOI: https://doi.org/10.1007/978-3-0348-0545-2_12
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